Nonintersecting Ryser Hypergraphs

Bishnoi, Anurag; Pepe, Valentina

doi:10.1137/19m1241556

Cited by 4 publications

(6 citation statements)

References 13 publications

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“…It was shown in [36] that for r = 3, this is essentially the only such example. However, it was shown in [1] that for r = 4, there is an example which is different than two disjoint 4-colored complete graphs and a more general example was given in [13].…”

Section: Lower Boundsmentioning

confidence: 99%

Generalizations and Strengthenings of Ryser's Conjecture

DeBiasio¹,

Kamel²,

McCourt³

et al. 2021

Electron. J. Combin.

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Ryser's conjecture says that for every $r$-partite hypergraph $H$ with matching number $\nu(H)$, the vertex cover number is at most $(r-1)\nu(H)$. This far-reaching generalization of König's theorem is only known to be true for $r\leq 3$, or when $\nu(H)=1$ and $r\leq 5$. An equivalent formulation of Ryser's conjecture is that in every $r$-edge coloring of a graph $G$ with independence number $\alpha(G)$, there exists at most $(r-1)\alpha(G)$ monochromatic connected subgraphs which cover the vertex set of $G$. We make the case that this latter formulation of Ryser's conjecture naturally leads to a variety of stronger conjectures and generalizations to hypergraphs and multipartite graphs. Regarding these generalizations and strengthenings, we survey the known results, improving upon some, and we introduce a collection of new problems and results.

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Section: Lower Boundsmentioning

confidence: 99%

Generalizations and Strengthenings of Ryser's Conjecture

DeBiasio¹,

Kamel²,

McCourt³

et al. 2021

Electron. J. Combin.

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“…Now we can state our result for the regular setting. For example, for a prime power q and dimensions 1 ≤ k < n, the k-dimensional affine subspaces in F n q form a resolvable 2-(q n , q k , n−1 k−1 q ) design, 5 whose dual is therefore a tight construction for Corollary 3.5. 6 In fact, we have a rich and storied variety of extremal constructions, as the study of resolvable designs dates back to Kirkman's famous schoolgirl problem [23] from 1857, which asked for resolvable 2-(15, 3, 1) designs.…”

Section: Strictly T-intersecting Hypergraphsmentioning

confidence: 99%

“…Although recent constructions of Abu-Khazneh[1], for ν = 2 and r = 4, and Bishnoi and Pepe[5], for ν ≥ 2 and all r ≥ 4 with r − 1 a prime power, show that one cannot simply reduce the general case to the intersecting one.…”

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confidence: 99%

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Ryser's Conjecture for $t$-intersecting hypergraphs

Bishnoi¹,

Das²,

Morris³

et al. 2020

Preprint

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A well-known conjecture, often attributed to Ryser, states that the cover number of an r-partite r-uniform hypergraph is at most r − 1 times larger than its matching number. Despite considerable effort, particularly in the intersecting case, this conjecture remains wide open, motivating the pursuit of variants of the original conjecture. Recently, Bustamante and Stein and, independently, Király and Tóthmérész considered the problem under the assumption that the hypergraph is tintersecting, conjecturing that the cover number τ (H) of such a hypergraph H is at most r − t. In these papers, it was proven that the conjecture is true for r ≤ 4t − 1, but also that it need not be sharp; when r = 5 and t = 2, one has τ (H) ≤ 2.We extend these results in two directions. First, for all t ≥ 2 and r ≤ 3t − 1, we prove a tight upper bound on the cover number of these hypergraphs, showing that they in fact satisfy τ (H) ≤ ⌊(r − t)/2⌋ + 1. Second, we extend the range of t for which the conjecture is known to be true, showing that it holds for all r ≤ 36 7 t − 5. We also introduce several related variations on this theme. As a consequence of our tight bounds, we resolve the problem for k-wise t-intersecting hypergraphs, for all k ≥ 3 and t ≥ 1. We further give bounds on the cover numbers of strictly t-intersecting hypergraphs and the s-cover numbers of t-intersecting hypergraphs.

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“…It was shown in [34] that for r = 3, this is essentially the only such example. However, it was shown in [1] that for r = 4, there is an example which is different than two disjoint 4-colored complete graphs and a more general example was given in [13].…”

Section: Lower Boundsmentioning

confidence: 99%

Generalizations and strengthenings of Ryser's conjecture

DeBiasio¹,

Kamel²,

McCourt³

et al. 2020

Preprint

View full text Add to dashboard Cite

Ryser's conjecture says that for an r-partite hypergraph H with matching number ν(H), the vertex cover number is at most (r − 1)ν(H). This far reaching generalization of König's theorem is only known to be true for r ≤ 3, or α(G) = 1 and r ≤ 5. An equivalent formulation of Ryser's conjecture is that in every r-edge coloring of a graph G with independence number α(G), there exists at most (r − 1)α(G) monochromatic connected subgraphs which cover the vertex set of G.We make the case that this latter formulation of Ryser's conjecture naturally leads to a variety of stronger conjectures and generalizations to hypergraphs and multipartite graphs. In regards to these generalizations and strengthenings, we survey the known results, improving upon some, and we introduce a collection of new problems and results.

show abstract

Nonintersecting Ryser Hypergraphs

Cited by 4 publications

References 13 publications

Generalizations and Strengthenings of Ryser's Conjecture

Generalizations and Strengthenings of Ryser's Conjecture

Ryser's Conjecture for $t$-intersecting hypergraphs

Generalizations and strengthenings of Ryser's conjecture

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